On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow

TL;DR

This paper proves the equivalence between bounded entropy and noncollapsing conditions for ancient Ricci flow solutions with nonnegative curvature, confirming Perelman's assertion under bounded geometry assumptions.

Contribution

It establishes the converse implication that noncollapsing implies bounded entropy for ancient solutions with bounded nonnegative curvature operator, completing the equivalence.

Findings

Bounded entropy implies noncollapsing for ancient solutions.

Noncollapsing implies bounded entropy under bounded curvature.

Equality between asymptotic entropy and asymptotic reduced volume is proven.

Abstract

In our previous work we showed that for an ancient solution to the Ricci flow with nonnegative curvature operator, assuming bounded geometry on one time slice, bounded entropy implies noncollapsing on all scales. In this paper we prove the implication in the other direction, that for an ancient solution with bounded nonnegative curvature operator, noncollapsing implies bounded entropy. Hence we prove Perelman's assertion under the assumption of bounded geometry on one time slice. In particular, for ancient solutions of dimension three, we need only to assume bounded curvature. We also establish an equality between the asymptotic entropy and the asymptotic reduce volume, which is a result similar to Xu, where he assumes noncollapsing and the Type I curvature bound.